Statistics and its Applications: Platinum Jubilee Conference, Kolkata, India, December 2016

178 Pages·2018·2.868 MB·English

by  Asis Kumar Chattopadhyay, Gaurangadeb Chattopadhyay

Springer Proceedings in Mathematics & Statistics
Asis Kumar Chattopadhyay 
Editors
Gaurangadeb Chattopadhyay   
Statistics
and its
Applications
Platinum Jubilee Conference, Kolkata,
India, December 2016
Springer Proceedings in Mathematics & Statistics
Volume 244
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Asis Kumar Chattopadhyay
Gaurangadeb Chattopadhyay
Editors
Statistics and its Applications
Platinum Jubilee Conference, Kolkata, India,
December 2016
123
Editors
Asis Kumar Chattopadhyay GaurangadebChattopadhyay
Department ofStatistics Department ofStatistics
University of Calcutta University of Calcutta
Kolkata, West Bengal, India Kolkata, West Bengal, India
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-981-13-1222-9 ISBN978-981-13-1223-6 (eBook)
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Contents
Fragmentation of Young Massive Clusters: A Hybrid Monte Carlo
Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Abisa Sinha
A Study on DNA Sequence of Rice Using Scoring Matrix Method
and ANOVA Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Anamika Dutta and Kishore K. Das
Regressions Involving Circular Variables: An Overview . . . . . . . . . . . . 25
Sungsu Kim and Ashis SenGupta
On Construction of Prediction Interval for Weibull Distribution. . . . . . 35
Ramesh M. Mirajkar and Bhausaheb G. Kore
Combining High-Dimensional Classification and Multiple Hypotheses
Testing For the Analysis of Big Data in Genetics. . . . . . . . . . . . . . . . . . 47
Thorsten Dickhaus
The Quantile-Based Skew Logistic Distribution with Applications. . . . . 51
Tapan Kumar Chakrabarty and Dreamlee Sharma
A Note on Monte-Carlo Jackknife: McJack and Big Data. . . . . . . . . . . 75
Jiming Jiang
A Review of Exoplanets Detection Methods . . . . . . . . . . . . . . . . . . . . . . 79
G. Jogesh Babu
A Connection Between the Observed Best Prediction and the Fence
Model Selection Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Thuan Nguyen and Jiming Jiang
A New Approximation to the True Randomization-Based Design
Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Siegfried Gabler, Matthias Ganninger and Partha Lahiri
v
vi Contents
Confounded Factorial Design with Partial Balance and Orthogonal
Sub-Factorial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Madhura Mandal and Premadhis Das
Beyond the Bayes Factor, A New Bayesian Paradigm for Handling
Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Christian P. Robert
No Calculation When Observation Can Be Made . . . . . . . . . . . . . . . . . 139
Tommy Wright
Design Weighted Quadratic Inference Function Estimators
of Superpopulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Sumanta Adhya, Debanjan Bhattacharjee and Tathagata Banerjee
Detecting a Fake Coin of a Known Type . . . . . . . . . . . . . . . . . . . . . . . . 163
Jyotirmoy Sarkar and Bikas K. Sinha
Fragmentation of Young Massive
Clusters: A Hybrid Monte Carlo
Simulation Study
AbisaSinha
Abstract TostudythehierarchicalfragmentationprocedureinYoungMassiveClus-
ters,astochasticmodelhasbeendeveloped.Binaryfragmentsalongwithindividual
starsareprimarilystudiedinthiswork.Stellarmassesforindividualstarshavebeen
generatedfromtheunivariatetruncatedParetodistributionandthestellarmassesfor
binarystarshavebeengeneratedfromthetruncatedbi-variateSkewNormalDistri-
butionusingtheHamiltonianMonteCarlomethod.Theabovedistributionisused
by observing the fitted bi-variate distribution of masses of all type of binary stars
viz.visualbinaries,spectroscopicbinariesandeclipsingbinaries.Theresultingmass
spectrumcomputedatdifferentprojecteddistancesareobservedunderopacitylim-
itedfragmentationprocedureandtheydisplaysignatureofmasssegregationalong
thecoretoradius,whereasdegreeofsegregationbecomesreducedduetoinclusion
ofalltypeofbinaryfragmentsincomparisontoinclusionofeclipsingbinariesonly.
· ·
Keywords Initialmassfunction Binarystars Bivariateskewnormal
HybridMonteCarlo
1 Introduction
TheInitialMassFunction(IMF)offragmentedmassesofmolecularcloudsunder-
goinggravitationalcollapseisoffundamentalinterestinmanyfieldsofastronomy
and astrophysics. First observed by Salpeter (1955), the IMF is a power-law of
the form ξ = dN ∝m(cid:3), where m is the mass of a star and N is the number of
dlogm
starsinthemassrangelogm and(logm+dlogm).Hisworkfavoredanexponent
of(cid:3) ∼−1.35for0.4M(cid:4) ≤m ≤10M(cid:4).Kroupaetal.(1993)found(cid:3) ∼−1.3(i.e.
α ∼2.3)abovehalfasolarmass,butintroducedα ∼1.3between0.08M(cid:4)−0.5M(cid:4)
and α ∼0.3 below 0.08M(cid:4) by proposing the IMF to be of segmented power-law
form,whereα =1−(cid:3) inlinearmassunitsoftheform dN ∝m−α.Moremodern
dm
B
A.Sinha( )
DepartmentofStatistics,BethuneCollege,Kolkata,India
e-mail:[email protected]
©SpringerNatureSingaporePteLtd.2018 1
A.K.ChattopadhyayandG.Chattopadhyay(eds.),StatisticsanditsApplications,
SpringerProceedingsinMathematics&Statistics244,
https://doi.org/10.1007/978-981-13-1223-6_1
2 A.Sinha
IMFs(segmentedpower-laws)appearedintheliteratureinrecentyears(e.g.Chabrier
(2003)),buthereourmainintentionwastofindhowtheobservedmassdistribution
depends on the slope of the fundamental IMF. Also, we have studied the effect of
masssegregationinYoungMassiveClusters(YMC)asaresultofinclusionofbinary
fragments.Miloneetal.(2012)haveinvestigatedthebehaviorofbinaryfractionfor
theGlobularClusters(GC)ofMilkyWayandhavefoundthatmasssegregationis
smallerthanthefieldbinaries.Wehavemodeledtherandomfragmentationscenario
togetherwithlineofsighteffectasproposedbyChattopadhyayetal.(2011, 2016).
Starsaregenerallybornasbinaryormultiplesystemsandtheirbinarynatureisvisi-
bleonlyforasmallpercentage.Chattopadhyayetal.(2016)havestudiedthebinary
fragmentsforeclipsingbinarystarsonly.Here,wehavetriedtoincludealltypesof
binarystarsviz.visualbinaries(consistsoftwostars,usuallyofdifferentbrightness
and observed visually), spectroscopic binaries (consisting of a pair of stars where
thespectrallinesinthelightemittedfromeachstarshiftsfirsttowardstheblue,then
towards thered,aseach moves firsttowards theobserver, andthenaway fromthe
observer,duringitsmotionabouttheircommoncenterofmass,withtheperiodof
theircommonorbitandobservedbyperiodicchangesinspectrallines)andeclipsing
binaries(consistingofstarsinwhichtheorbitplaneofthetwostarsliessonearly
in the line of sight of the observer that the components undergo mutual eclipses
andobservedonlybystudyingtheirrespectiveLightCurvesandVelocityCurves).
Severalauthorshavestudiedthemasses,massratiosofthesebinarystars(Tokovinin
2014;Kouwenhovenetal.2007).
Thepercentageofbinarycontributiontothefinalformoffragmentsisofconsid-
erable debate. Over the past few years, several authors have considered the binary
fragmentsinacoventionalway(Abt1983forBstars;DuquennoyandMayor(1991)
for G dwarfs; Fischer and Marcy (1992) for M dwarfs; Kouwenhoven et al. 2007
forAandBstars;Goodwinetal.(2007)andreferencestherein).Similarobserva-
tionswerenotedfornearbyandassociatedclusters(Duchéne1999;Duchéneetal.
2007).Binaryfractionsofdistantclusterswerenotobservedpreviouslybecauseof
observationallimitationsofmeasuringdevices.Fortunately,bysomealternativetech-
nique,someauthorshavebecomeabletocalculatethebinaryfraction.Forexample,
by studying the morphology of colour-magnitude diagram, Romani and Weinberg
(1991)determinedtheobservedbinaryfractionsin M92and M30at≤9%and4%
respectively. Rubenstein and Bailyn (1997) investigated binary fraction of stars in
therange15.8mag<V<28.4magin13.5GyroldGalacticGC,NGC6752as15–
38%insidetheinnercore,fallingto16%atlargerradiiwithapower-lawmassratio
distribution.Ballazzinetal.(2002)estimatedthebinaryfractioninNGC288forstars
20mag<V<23mag(∼0.54–0.77 M(cid:4))as8–38%insideclusterhalfmassradius.
ZahoandBailyn(2005)found6–22%ofmainsequencebinariesforM3withincore
radiuswhereasCoolandBolten(2002)derivedabinaryfractionof3%forGalactic
GC,NGC6397.RomaniandWeinberg(1991)andHurleyetal.(2007)estimateda
binaryfraction5.1±1.0%withintheinnerregionofNGC6397.Alltheclustersare
dynamicallyevolvedsystemsandareexpectedtosignificantlyaltertheinitialbinary
population.Huetal.(2010)havestudiedtheyoungstarclusterNGC1818inLMC
(age∼15–25Myr)andderivedabinaryfractionashighas55%.Chattopadhyayetal.
FragmentationofYoungMassiveClusters:AHybridMonteCarloSimulationStudy 3
(2016)hasconsideredthecontributionofbinaryfragmentsascloseas50%,while
consideringtherestassinglestars.MalkovandZinnecker(2001)haveevenclaimed
thatthecontributionofbinaryfragmentsisascloseto100%.
In the present work, we have considered random fragmentation of YMCs and
havetakenthebinarycontributiontobeof80%ofthetotalfragmentswhereas20%
constitutessinglestars.Wehavesimulated80%ofbinarystarsfromthetruncated
Bi-variate Skew Normal Distribution by Hybrid Monte Carlo method and the rest
20% of single stars from truncated Pareto distribution, truncated at minimum and
maximummasses.Thepatternofthebi-variatedistributionisinvestigatedandfitted
to an appropriate form. In Sect.2 we have discussed the data set, Sect.3 gives the
form of bivariate distribution, Sect.4 gives the simulation procedures and Sect.5
givestheresultsanddiscussions.
2 DataSetofBinaryStars
Wehaveusedadata-setof2096binarystarscomprisingofvisualbinaries,spectro-
scopicbinariesandeclipsingbinaries,amongwhich1875setsofmassesaretaken
fromTokovinin(2014)constitutingonlythosebinarystarswhichmaybeobserved
through telescope (viz. visual binaries, spectroscopic binaries), 78 sets of masses
takenfromKouwenhovenetal.(2007)(alsoconstitutingvisuallyobservablestars)
and the rest 143 sets of masses of eclipsing binaries (observed from their light-
curvesandvelocity-curves)fromChattopadhyayetal.(2016).Themethodusedto
calculatebinarymassesfromtheirobservedmass-ratiosasinTokovinin(2014)and
Kouwenhovenetal.(2007)hasbeenfoundcompatibleforuse(Fig.3).
3 Bi-variateDistribution
3.1 Distribution Fit
Initially,wedisplayedthedataofbinarymasses,inabi-variateplot(Bivariatehis-
togram(Fig.1))whichdisplaysapositiveskewedpattern.Theabovedatasetisthen
fittedtobi-variateskewnormaldistributionoftheform:
fZ1,Z2(z1,z2)=2φ2(z−ξ;(cid:6))(cid:7)(α(cid:6)ω−1(z−ξ)) (1)
wherez ,z aretherandomvariablesrepresentingthemassesofbinarystarsm ,m
1 2 1 2
rveaslpueecstoivfeZly,,fZZ1,Zr2e(szp1e,czt2iv)eislyt.hHeeprreobξab=ili(cid:2)tξydξen(cid:3)s(cid:6)iitsytfhuencloticoantiaonndpza1r,azm2aerteerp,a(cid:6)rtiicsutlhaer
1 2 1 2 (cid:2) (cid:3)
correlation matrix (technically known as scale parameter) and α = α α (cid:6) is the
1 2
shapeparameterwhichneedstobeestimated.